Properties

Label 720.4.a.f
Level $720$
Weight $4$
Character orbit 720.a
Self dual yes
Analytic conductor $42.481$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 720.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(42.4813752041\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 120)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 5 q^{5} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 5 q^{5} - 8 q^{7} + 20 q^{11} + 22 q^{13} + 14 q^{17} - 76 q^{19} + 56 q^{23} + 25 q^{25} + 154 q^{29} - 160 q^{31} + 40 q^{35} - 162 q^{37} + 390 q^{41} - 388 q^{43} - 544 q^{47} - 279 q^{49} + 210 q^{53} - 100 q^{55} - 380 q^{59} - 794 q^{61} - 110 q^{65} + 148 q^{67} - 840 q^{71} + 858 q^{73} - 160 q^{77} - 144 q^{79} + 316 q^{83} - 70 q^{85} - 1098 q^{89} - 176 q^{91} + 380 q^{95} + 994 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 0 0 −5.00000 0 −8.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(5\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 720.4.a.f 1
3.b odd 2 1 240.4.a.d 1
4.b odd 2 1 360.4.a.e 1
12.b even 2 1 120.4.a.f 1
15.d odd 2 1 1200.4.a.bf 1
15.e even 4 2 1200.4.f.g 2
20.d odd 2 1 1800.4.a.k 1
20.e even 4 2 1800.4.f.h 2
24.f even 2 1 960.4.a.g 1
24.h odd 2 1 960.4.a.v 1
60.h even 2 1 600.4.a.d 1
60.l odd 4 2 600.4.f.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.4.a.f 1 12.b even 2 1
240.4.a.d 1 3.b odd 2 1
360.4.a.e 1 4.b odd 2 1
600.4.a.d 1 60.h even 2 1
600.4.f.g 2 60.l odd 4 2
720.4.a.f 1 1.a even 1 1 trivial
960.4.a.g 1 24.f even 2 1
960.4.a.v 1 24.h odd 2 1
1200.4.a.bf 1 15.d odd 2 1
1200.4.f.g 2 15.e even 4 2
1800.4.a.k 1 20.d odd 2 1
1800.4.f.h 2 20.e even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(720))\):

\( T_{7} + 8 \) Copy content Toggle raw display
\( T_{11} - 20 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T + 5 \) Copy content Toggle raw display
$7$ \( T + 8 \) Copy content Toggle raw display
$11$ \( T - 20 \) Copy content Toggle raw display
$13$ \( T - 22 \) Copy content Toggle raw display
$17$ \( T - 14 \) Copy content Toggle raw display
$19$ \( T + 76 \) Copy content Toggle raw display
$23$ \( T - 56 \) Copy content Toggle raw display
$29$ \( T - 154 \) Copy content Toggle raw display
$31$ \( T + 160 \) Copy content Toggle raw display
$37$ \( T + 162 \) Copy content Toggle raw display
$41$ \( T - 390 \) Copy content Toggle raw display
$43$ \( T + 388 \) Copy content Toggle raw display
$47$ \( T + 544 \) Copy content Toggle raw display
$53$ \( T - 210 \) Copy content Toggle raw display
$59$ \( T + 380 \) Copy content Toggle raw display
$61$ \( T + 794 \) Copy content Toggle raw display
$67$ \( T - 148 \) Copy content Toggle raw display
$71$ \( T + 840 \) Copy content Toggle raw display
$73$ \( T - 858 \) Copy content Toggle raw display
$79$ \( T + 144 \) Copy content Toggle raw display
$83$ \( T - 316 \) Copy content Toggle raw display
$89$ \( T + 1098 \) Copy content Toggle raw display
$97$ \( T - 994 \) Copy content Toggle raw display
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